It is well known that in Descriptive Complexity Theory FO is equivalent to AC0.

However, this accepts a couple of a theory and a string <T,s> iff the interpretation of s as a query is satisfied by the theory T.

My question now is if there is an interpretation of which is stronger than AC0. For example, is there an interpretation of s as a theory so that the formalism accepts if s is a theory that is logically entailed by T. This problem is undecidable, but maybe it is possible to express problems in it which are stronger than AC0. (Reachability would be an interesting example if anyone could make this work)



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